Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MML1004	Commutative Rings and Number Theory	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	3	2+1+0	speciality	optional	
Didactic Mathematics - in Hungarian	3	2+1+0	speciality	optional	
Interdiciplinary Computational - in Hungarian	3	2+1+0	speciality	optional	

Teaching Staff in Charge	Prof. MARCUS Andrei, Ph.D., marcus@math.ubbcluj.ro	
Aims	Deepening the knowledge on Number Theory from a higher point of view.
Introduction to Algebraic Number Theory. Developing problem solving skills.

Content	1) Divisibility in integral domains. 
       1.1. Divisibility in Z. 
       1.2. Prime numbers
       1.3. Arithmetic functions
       1.4. Factorial rings
2) Congruences 
       2.1. Sistems of linear congruences. The Chinese Remainder Theorem
       2.2. The group of units of Z_n
       2.3. n-th power residues
3) Quadratic residues
       3.1. The Legendre symbol
       3.2. Quadratic reciprocity
       3.3. The Jacobi symbol
4) Quadratic fields and rings of quadratic integers 
       4.1. Ramification
       4.2. Euclidean rings
5) Diophantine equations (1)  
       5.1. Equations of degree 1
       5.2. Pithagorean numbers
6) Diophantine equations (2)  
       6.1. About Fermat@s Last Theorem
       6.2. Case n=4  
       6.3. Case n=3  
7) Diophantine equations (3)  
       7.1. Pell@s equation 
8) Commutative rings  
       8.1. Noetherian rings and modules
       8.2. Rings and modules of fractions
9) Fields of algebraic numbers  
       9.1. Algebraic extensions of fields
       9.2. Cyclotomic fields
10) Rings of algebraic integers (1)  
      10.1. Algebraic integers
      10.2. Trace, norm, discriminant
11) Rings of algebraic integers (2)  
      11.1. Unique factorization
      11.2  Ramification ans degree      
12) Valuation rings
      12.1. Valued fields
      12.2. Discrete valuation rings
13) Dedekind rings 
      13.1. Definition
      13.2. Characterizations
14) Rings of algebraic integers (3) 
      14.1. Cyclotomic fields: intregi si ramificare 
      14.2. The Kronecker-Weber  theorem
      14.3. The group of units of a ring of algebraic integers 
	
Bibliography	1. K. Ireland, M. Rosen, A Classical Introduction to Modern Number Theory, Springer Verlag Berlin 1990
2. T.Albu, Ion D. Ion, Capitole de teoria algebrica a numerelor, Editura Academiei,  Bucuresti, 1984
3. Lang S., Algebra, Springer Verlag Berlin, 2002
4. ROTMAN, J.: Advanced modern algebra, Prentice Hall, NJ 2002
5. A. MARCUS: Algebra [http://math.ubbcluj.ro/~marcus]

Assessment	Homeworks (20%). Exam. (80%)